Does the surface area change if the arrangement (configuration) of cubes
changes?

Problem 1 How many cubes are needed for the cuboid
opposite?

Problem 2 Find the surface area of this cuboid.

Problem 3 Using 24 cubes to make a cuboid, what is the
configuration that gives

a)minimum surface area

b)maximum surface area?

The minimum surface area configuration corresponds to the shape that
requires the minimum amount of wrapping paper for a given volume (i.e. 24
cubic units). Of course, overlaps would be needed in practice, but
as they would be similar for all shapes, we can disregard them hee.

Problem 4 Find the minimum wrapping
for

a) 48 cubes
b) 64 cubes.

A related problem is to find the total amount of string required to go round
the cuboid in each direction (as shown opposite), not including the
extra needed for the knot.

Problem 5 For the cuboids you made earlier from 24 cubes, find
the total length of string needed for each configuration.

Problem 6 What configuration minimises the amount of string
needed to tie up a parcel of volume 24 cubic units?

EXTENSION

1. Find the cuboid shape that minimises the surface area when
it encloses a fixed volume, V.